Step 1: Factor out $A^{2023}$ We can factor out $A^{2023}$ from the expression inside the determinant: $|A^{2025} - 3A^{2024} - A^{2023}| = |A^{2023}(A^2 - 3A - I)|$

Step 2: Use determinant properties Using the property $|AB| = |A||B|$, we have: $|A^{2023}(A^2 - 3A - I)| = |A^{2023}| |A^2 - 3A - I| = |A|^{2023} |A^2 - 3A - I|$

Step 3: Calculate $|A|$ $|A| = (2)(5) - (3)(3) = 10 - 9 = 1$

Step 4: Calculate $A^2$ $A^2 = \begin{bmatrix}2&3\\3&5\end{bmatrix} \begin{bmatrix}2&3\\3&5\end{bmatrix} = \begin{bmatrix}4+9&6+15\\6+15&9+25\end{bmatrix} = \begin{bmatrix}13&21\\21&34\end{bmatrix}$

Step 5: Calculate $A^2 - 3A - I$ $A^2 - 3A - I = \begin{bmatrix}13&21\\21&34\end{bmatrix} - 3\begin{bmatrix}2&3\\3&5\end{bmatrix} - \begin{bmatrix}1&0\\0&1\end{bmatrix} = \begin{bmatrix}13&21\\21&34\end{bmatrix} - \begin{bmatrix}6&9\\9&15\end{bmatrix} - \begin{bmatrix}1&0\\0&1\end{bmatrix} = \begin{bmatrix}13-6-1&21-9-0\\21-9-0&34-15-1\end{bmatrix} = \begin{bmatrix}6&12\\12&18\end{bmatrix}$

Step 6: Calculate $|A^2 - 3A - I|$ $|A^2 - 3A - I| = (6)(18) - (12)(12) = 108 - 144 = -36$

Step 7: Calculate $|A|^{2023} |A^2 - 3A - I|$ $|A|^{2023} |A^2 - 3A - I| = (1)^{2023} (-36) = 1 \cdot (-36) = -36$

Step 8: Take the absolute value Since the question asks for the absolute value, we have: $|-36| = 36$